This paper investigates the role of foundational theories in the construction of formal ontologies. Whereas current practical ontologies employ foundational theories primarily to organise the upper-level categories of their class hierarchy, this paper explores an approach that builds upwards from fundamental concepts of space, time, and matter. The paper explores the foundations of a definitional methodology in which all concepts are firmly grounded upon basic primitives and the vocabulary is extended incrementally towards more human-oriented, mid-level vocabularies. A significant advantage of this approach is that when new vocabulary is introduced, no additional axioms are required beyond the definitions themselves: semantic constraints holding between defined concepts are automatically enforced by the axioms governing the primitive terms. The paper highlights some meta-logical concepts that can help identify definability relationships among terms within an ontology. The gap between foundational theories and more general conceptual terms is acknowledged and ways to bridge this are explored. In particular, the paper considers the nature of matter and how this can be introduced into a foundational theory. In addressing this, the paper examines a puzzle, first identified by Tarski, regarding the axiomatisation of a theory of matter and mechanics within any framework that incorporates a theory of geometry. A solution to the puzzle is proposed in which a primitive concept of 'matter' acts as both a symmetry breaking reference frame that conceptually enriches a purely spatial ontology, and also as a representation of the actual material substance of the world.
Since ancient times, a central goal of those investigating the nature of reality has been to determine the fundamental elements of which it is composed, and how the rich and varied structure of the world we observe is constructed from those elements. This goal has been pursued both in relation to the material constituents of the world (culminating in atomic and sub-atomic physics) and also its conceptual elements, the basic ontological categories that underlie our apprehension of the world in thought and language.
Scientific theories have been developed at several diferent levels of scale, and in relation to diferent types of element and diferent modes of behaviour. Hence, we have atomic and sub-atomic physics, chemistry, mechanics, cosmology, biology, psychology etc. But, notwithstanding this division into sub-disciplines, our modern conception of science is that it must regard all aspects of the world as operating according to a single set of fundamental principles, and moreover that structure and behaviour observable at large scales is completely determined by the behaviour of smaller scale elements of reality. Indeed, the interactions between diferent fields of science (such as between microscopic and macroscopic physics, chemistry and biology, biology and psychology etc.) are areas of investigation that have resulted in theories providing bridges between them (e.g. we have well-developed accounts of how atomic physics explains chemical interactions).
The person most responsible for the conception of ontology and who has had a strong and lasting influence on its development is Aristotle. 1 His method of analysis particularly emphasised the species, in other words natural kinds of things, as being the level of categorisation that most clearly reveals the essential characteristics of things (i.e. the properties that are intrinsic to their nature rather than just a result of particular circumstances).2 Starting with a loose idea of similar things forming a species, one can define species more precisely by: first, considering broader categories, genera, that incorporate several species; and then, identifying diferentia, which are the salient properties that divide a genus up into its separate species.
The Aristotelian approach investigates and reveals a hierarchical structure, or taxonomy, that classifies
elements of reality into groups and describes the logical relationships between these groupings and
the properties that give rise to these relationships. Certainly, this kind of structure and the logic that
it captures are useful for many tasks involving manipulation of information expressed by means of
conceptualisations based on those used in human thought and language. Because of this, it plays a core
role in the design of modern ontologies that are used for computational processing of information to
support human activities. The extent to which modern computation-oriented ontologies are developed
according to Aristotelian principles varies. The stated methodology of BFO explicitly acknowledges its
Aristotelian influence [
Aristotle’s approach to describing reality can be contrasted with that of Democritus (and subsequently Epicurus, who held a similar view). Democritus was more interested in the physical and material properties of the world. He proposed the existence of atomic particles that were the ultimate constituents of reality and considered large-scale physical objects and their behaviour to be completely determined by the properties of their microscopic atomic constituents. It is clear that the Democritean conception of physical reality is closely aligned to, indeed prophetic of, the theories of modern physics. Nevertheless, with regard to conceptual analysis of reality by philosophers through most of history, and right up to present day ontology development, the species-based approach of Aristotle has eclipsed the atom-based approach of Democritus.
But there are exceptions to this. Perhaps most prominent is the methodology of Carnap, as propounded
in his influential book Der logische Aufbau der Welt [
The Carnapian project was extremely influential and was the major force behind the Logical Positivist movement in philosophy during the first half of the 20th century. However, by the latter half of that century, its influence had declined due to technical dificulties, lack of obvious applications and the move towards a view of language as being too flexible and pragmatic to be amenable to description in formal logic. Nevertheless, now that the value of ontologies in the fields of computation and information science is widely recognised, and the challenges of constructing such ontologies are accepted as being very large but worth tackling, the possibility of constructing strongly definitional ontologies may be more plausible and attractive.
The reductivist approach to ontology that I am considering is related to but distinct from the notion
of reductionism that is often debated by philosophers of science (e.g. [
What is the role of foundational theories in the construction of ontologies? The obvious answer would
be that they provide a foundation upon which the rest of the ontology is built. Yet it seems that typically
this is not the case. In the development and use of practical ontologies that are actually employed
within information systems, foundational theories (such as theories of space and time) mostly play
a more peripheral, albeit important role. They typically provide top-level categories used to classify
entities into distinct ontological kinds and correspondingly distinguish lower-level categories according
to the ontological type of their instances. For example, the Cell Ontology and Gene Ontology use BFO
[
One reason why a grounded approach has not been adopted is that it is not necessary for many applications that are limited to specific domains. Purely taxonomic reasoning is very useful and can be captured by general relations between species without one needing to ground the meanings of species concepts. And when the diferentia underpinning the taxonomy are also represented as explicit combinations of properties and relationships, the system becomes more powerful still.
Yet, the logic of taxonomies is only one aspect of human reasoning. To understand a sentence such as: “Sue often played with her friend Tom, but Tom’s father owned two Rottweilers, and Sue was afraid of dogs, so Sue never visited Tom’s house.”, we do use taxonomic knowledge to recognise that Rottweiler is a type of dog; however, we also need detailed knowledge about ownership, domestic arrangements and human psychology that goes well beyond purely taxonomic information. And, to understand “The boat sank because it was loaded with too much coal.” one needs knowledge of commonsense physics for which taxonomic relationships have little relevance.
Traditionally, knowledge needed to interpret such examples has been captured by augmenting the ontology with extensive axiom sets encoding commonsense background knowledge, as advocated by the Naïve Physics Manifesto [22] and exemplified by the CYC project [ 23]. An alternative, however, is to investigate how such knowledge might emerge from more fundamental conditions of reality, with higher-level concepts introduced incrementally by means of definitions grounded in those underlying structures. This latter approach may initially seem implausible, since high-level, human-oriented notions—such as being afraid of dogs—appear to be so far removed from fundamental concepts like space and time that it is dificult to conceive how they could be grounded in such primitives. And yet, a complete specification of the unfolding material states of the universe would surely determine not only all occurrences of boat-sinking events, but also all situations in which human fear of dogs is manifest—not only in actual encounters with dogs, but also in conversations, physiological responses such as muscular tension, or avoidance behaviours concerning dogs. The fact that such high-level phenomena are, in principle, fully determined by the underlying succession of physical states lends support to the view that they might ultimately be definable in terms of basic primitives.
The implausibility of actually formulating adequate definitions may be attributed to the apparent incompatibility between diferent levels of description—for example, between physical theories framed in terms of particles and forces, and the human-oriented perspective structured around physical objects and their afordances, or between material accounts of the world and those expressed in terms of beliefs, desires, and intentions. This apparent detachment of high-level, human-oriented concepts from material primitives—along with the presence of many evident semantic relationships at that level (such as “people avoid what they are afraid of”)—has understandably led those constructing AI knowledge bases aimed at reproducing human reasoning to state axioms directly in terms of high-level concepts, rather than attempt the dificult task of grounding them in more fundamental terms.
In this paper, I take an alternative approach: rather than positing high-level concepts directly, I investigate how they might be built up in definitional layers from fundamental primitives. This strategy may be viable if we proceed incrementally and are able to bridge certain dificult conceptual gaps along the way. The focus will be on the foundations—specifically, how to introduce the notion of matter into a formal theory of space and time, and how this can serve as a basis for developing a theory of physical objects.
It should be noted that other approaches to grounding can be taken such as the constructivism of the Erlangen School [24, 25], which feed into more recent ideas for ontology developement and validation, such as proposed in [26]. These make diferent choices regarding what primitives should provide the grounding, which may be advantageous for formulating ontologies concerned with human-oriented conceptualisations.
We now give an overview of definitions from a formal perspective and highlight some significant meta-logical properties and theorems. For current purposes, we confine attention to theories expressed in standard first-order logic. However, most of the notions introduced would apply to any of a broad class of logical languages.
Definition 2.1. 1st-order definitional formulae take one of the following forms, depending on the type of symbol defined: • • • ∀1 . . . [(1, . . . , ) ↔ Ψ( 1, . . . , )] ∀1 . . . [( = (1, . . . , )) ↔ Ψ(, 1, . . . , )] ∀[( = ) ↔ Ψ()]
The definienda (terms defined) by such formulae are respectively: the relation the constant . The expression Ψ( 1, . . . , ) is called the definiens.
A first-order definition is any formula of one of the above forms in which the definiendum does not occur in the definiens. , the function and
The notion of definability allows us to identify subsets of the vocabulary of a theory which can be taken as ‘primitive’ concepts: Definition 2.2. A primitive vocabulary for a theory Θ is a set ⊆ Voc(Θ) Voc(Θ) are definable within Θ from the symbols in . such that all symbols in Definition 2.3. A minimal primitive vocabulary for a theory Θ is a primitive vocabulary for Θ of which no proper subset is also a primitive vocabulary for Θ.
Within the context of a formal theory it is natural to consider definitions and definability from a syntactic point of view. However, one can also gain considerable insight into these notions by considering how possible dentotations of concepts are related within the possible models of a theory.
Padoa’s Method [27] provides a way to tell whether some term (i.e. a predicate or relation, symbol or a constant name) employed within in an axiomatic theory is independent of the others, or on the contrary is definable in terms of one or more of the remaining terms. In terms of the modern parlance of model theory, Padoa’s key notion of independence can be specified as follows: Definition 2.4. Suppose we have an axiomatic theory Θ that makes use of a vocabulary of non-logical terms = {1, 2, . . . , }, with ∈ and ⊆ ∖{} . Then the term is independent from terms just in case we can find two models that satisfy all the axioms of Θ and agree exactly on the denotations of all the symbols in but disagree on the denotation of . If there are no such models then is dependent on , in the context of Θ.
It is readily apparent that being dependent on within Θ is a necessary condition for being definable from the terms in Θ . If not then any proposed ‘definition’ of that fixed its interpretation in terms of the symbols would rule out some possible models of Θ , and hence would not be purely definitional.
Whether the converse is true, that dependence is suficient for definability, is not so clear. However, Tarski, in [28, 29] showed that dependence is both necessary and suficient for definability in higher logic; and, subsequently, Beth [30] proved with his famous definability theorem that, if we are dealing with a first-order theory, we can be sure that any dependent symbol can be given a first-order definition: 3 3Beth refers to Padoa’s notion of dependence as implicit definability, and the existence of a first-order definition is called explicit definability. In this nomenclature, the theorem states that implicit and explicit definability coincide for first-order theories. Theorem 1. For every first-order theory Θ , a symbol is dependent on a set of symbols within Θ just in case Θ entails a first-order definition formula, whose definiendum is and with only non-logical symbols from occurring in its definiendum.
We can illustrate this with a concrete example. Suppose we have the following set of human property and family relationship terms (where the notation Pred/ indicates that predicate Pred has arity ): {Male/1, Female/, IsParentOf/2, IsMotherOf/2, IsMarriedTo/2, IsMotherInLawOf/2} It should be clear that {Male/1, Female/, IsParentOf/2, IsMarriedTo/2} provide a suficient basis of terms to define the other two concepts. Moreover, if we had an axiom ∀[Male() ↔ ¬Female()], then we only need one of the two gender predicates. We also see that we could define IsMotherOf/2 from Female/2 and IsParentOf/2 without needing the IsMarriedTo/2 predicate.
We could show these dependencies either by considering which definitions would hold in a reasonable axiomatisation of these predicates or we can use Padoa’s method. To use this method we just need to recognise that for any model — i.e. in this case a collection of humans for whom certain family relations hold. We can then see immediately that if we fix the only gender and parenthood relation then IsMotherOf will be fixed, but since IsMarriedTo is not fixed then IsMotherInLawOf is also not fixed. So the latter cannot be defined without using the former.
In many areas of mathematics and science, principles similar in form and purpose to Padoa’s method are widely employed whenever one is interested in establishing what causal factors that are relevant to a particular efect. We may think that we have identified all factors relevant to that efect since, in cases that are equivalent in terms of these factors, the efect has been found to be the same. However, if we later find two cases that are equivalent with respect to those factors but difer in the efect observed, we must accept that there must be other relevant factors.
The investigations of Tarski in his paper Some methodological investigations on the definability of concepts [28, 29] proved that the definability relationships among the terms of a theory is completely determined by certain properties concerning isomorphisms between models of that theory.
Definition 2.5. An isomorphism between two models ℳ and (with the same signature) is a bijective mapping between their domains that preserves and reflects the interpretation of all non-logical symbols. That is, relations, functions, and constants are mapped identically under the correspondence.
A theory is categorical if and only if there is at least one isomorphism between any two Definition 2.6. models.
Definition 2.7. A theory is monotransformable if and only if given any two models, there is at most one isomorphism between them.
Definition 2.8. A theory Θ is conceptually complete4 if there is no categorical theory Θ ′ such that Θ ⊂ Θ ′ and Voc(Θ ′) contains at least one symbol that is not definable within Θ ′ in terms of the vocabulary Voc(Θ).
So a ‘conceptually complete’ theory is one of which no categorical extension contains any new concept that is not definable (in the extended theory) from the vocabulary already present in the original theory. Note that if we dropped the restriction of considering only categorical extensions then no theory would be conceptually complete: it is always possible to add extra undefinable terms to a theory if we don’t care about exactly specifying their meaning.
Though conceptual completeness is concerned with definability in categorical extensions of a theory
it is still a useful notion for theories that we do not intend to axiomatise up to categoricity. Given
that any categorical axiomatisation of a new concept within a conceptually complete theory would
entail a definition, it will normally be more sensible to define the concept rather than give a partial
axiomatisation. Alternatively, if we do want to explicitly ensure categoricity with respect to an ontology
(or perhaps to a subdomain of an ontology) we would need to use an ontology with a language richer
than first-order logic, such as the General Formal Ontology (GFO) [
It is worth noting that conceptual completeness of a theory does not mean that it is capable of defining all possible relationships over its domain. Indeed, if the domain is infinite there will be an uncountable number of possible predicate extensions, so these cannot all be defined within a language with a recursively enumerable set of well defined formulae. The property of conceptual completeness just means that the definitional power of the theory cannot be extended by adding more primitives.
Now we come to the key theorem proved in [28, 29] where Tarski showed that the model-theoretic property of monotransformability corresponds exactly to conceptual completeness: Theorem 2. A theory is conceptually complete if and only if it is monotransformable.
This tells us that conceptual completeness amounts to the condition that no model of the theory incorporates multiple identical copies of itself. This is because if such copies existed, we could add concepts that would distinguish between copies.
A good example of a theory that is not monotransformable, but can be made so, is the theory of a strict discrete total order relation such that for every element there is both a greater and a smaller element. All models of such a theory will be isomorphic to the integers. However, we can map any model to itself in such a way that the order is preserved but each element gets mapped to a diferent element at some fixed shift up or down the order structure (e.g. we can map each element to its successor). But if we designate one element by a special name, say ‘0’, no such shift is possible without changing the denotation of ‘0’, so the theory becomes monotransformable. Hence, the theory is also conceptually complete. No concept can be added to the theory that is not already definable.
The following consequence of Theorem 2 (also given in [28, 29]) is easy to comprehend, when we consider that once we add a fixed coordinate system to space, the only automorphism that preserves that coordinate system is the identity mapping: Theorem 3. Euclidean geometry with a fixed coordinate system stipulated by additional constants is monotransformable, and hence conceptually complete.
It is likely that anyone with experience of constructing ontologies to support functionality of computational information systems, will consider that analysis of definability in terms of symmetries of categorical models can have no direct bearing on practical ontology design. They might say that ‘real’ ontologies cannot even avoid non-intended models, let alone ensure categoricity or monotransformability, so consideration of such abstruse meta-semantic properties cannot provide any useful guidance. I think this would be a misapprehension for the following reasons:
Firstly, the Method of Padoa can certainly be used in an informal and useful way. When considering vocabulary of related properties and relations it is surely useful to consider whether the interpretation of some subset of the vocabulary would fix the interpretation of the remaining terms. Then one could concentrate on axiomatising the subset (the primitives) while stipulating the meanings of the remaining terms by purely definitional axioms.
Secondly, although categoricity is not a property that one would expect in practical ontologies, this condition is not required for conceptual completeness. Monotransformability, is the key concept and implies that all terms would be definable if the theory were to be extended to a categorical theory, we do not actually need to do this. In fact for most ‘ordinary’ ontologies extending to a categorical theory is possible in principle. We would need to specify all the objects and the primitive (i.e. undefinable) properties and relations that hold. Although impractical, this must be possible for any consistent theory that can have a finite domain (e.g. of physical objects). It would also apply to theories that have domains, or sub-domains that are infinite but can be categorically axiomatised (e.g. a time series and/or spatial geometry).
(a) (b) (c)
We cannot articulate our commonsense conceptualisation of the world in terms of vocabulary that describes only purely spatial and temporal aspects of reality. There needs to be some stuf in space to constitute physical objects; and the stuf needs to move around in order to manifest the processes and events that occur as the world evolves through time. Moreover, there does not seem to be any possibility that physical objects and their properties could be defined purely in terms of space and time, at least not in terms of conventional theories of space and time.
In the Newtonian theory, space and time characterise the structure of a void within which matter exists. They do not provide any properties that could determine the location of a particular kind of matter at a particular place in a particular time. And in terms of the Method of Padoa, we see that all purely spatial and temporal relationships could stay the same while the relationships pertaining to configurations of matter can vary. Of course matter is located in space, but space itself does not determine the location of matter.
The nature of foundational theories is that they describe the structure of reality in a way that is so general that they do not distinguish any specific features of the particular reality in which we live. Purely spatial, temporal and spatio-temporal relationships may apply to arbitrary regions of space (e.g. a region may be spherical, or overlap another region, or a spatio-temporal volume may correspond to a continuous path in space-time); but they do not pick out particular regions as being, say, occupied by a lump of stone, or corresponding to the trajectory of a football.
These considerations suggest that for a theory to support definitions of properties and relationships between physical objects it must include some primitives that are not purely spatio-temporal in character. But Tarski’s result regarding conceptual completeness of Euclidean geometry in [29] implies that any theory that includes Euclidean geometry can be formulated purely in terms of geometrical primitives. As Tarski notes in the conclusion to [29], this gives rise to a puzzle with regard to theories that describe the properties and behaviour of physical objects in space (e.g. Newtonian mechanics). On the one hand, it seems that we must need more than purely spatial concepts to capture matter and its mechanics; but on the other hand, the conceptual completeness of Euclidean geometry means that any further concepts added to the theory would in fact be definable from the original primitives.
Tarski suggested that the situation might indicate that formulation of an adequate physical theory requires a radically diferent kind of geometry. This may indeed be so. If instead we consider space and matter in the context of Einstein’s General Relativity, we find a strong interdependence between ‘spacetime’ and matter. Yet neither is fully dependent on the other and it is not possible to eliminate matter from its equations. We should note that there are more exotic theories (e.g. Loop Quantum Gravity), where matter does indeed arise from the structure of space time itself. However, these possibilities do not really explain the situation with regard to Euclidean geometry and classical mechanics. Surely it is possible to have a consistent classical mechanics incorporating standard geometry ([32] and [33] seem to demonstrate this). So the question of how to define matter within such a theory remains.
I suggest that the solution to this puzzle is that the material content of the world can be introduced into the theory at the point where the coordinate system is fixed. As we saw above, achieving definitional completeness requires that a theory be made monotransformable. To do this we need to eliminate isomorphic copies within the structure, so that the only automorphism on the domain is the identity mapping. In the case of geometrical theories, this would normally be done by introducing some specific points or regions to act as an anchor for a coordinate system. But there is no reason why we should only introduce a few named spheres, as in Fig. 1(a). We can in fact introduce any designation whose structure is suficiently complex to break all symmetries of empty space.
There are many ways in which this reference structure could be specified. But there is one way that enables us (despite Theorem 3) to enrich the theory with a concept of matter and extends its conceptual repertoire beyond that of purely spatial properties and relations. To do this we designate a particular sphere by a constant ‘unit’, and take its diameter as a unit of length; and, more significantly, we also introduce another constant ‘matter’, which has a dual purpose in the theory. One purpose is that ‘matter’ designates a region which we use to anchor the reference frame. But, as you may have guessed, we can also interpret ‘matter’ to be the extent of physical matter as it is distributed in space.
In order that the new primitives ‘unit’ and ‘matter’ fulfil their purposes we need to specify that they satisfy some axiom of the form:
Sphere(unit) ∧ Φ(matter, unit) , where Φ describes a particular shape in space occupied by ‘matter’, whose size is calibrated by ‘unit’.
In order for Φ to be suitable for picking out a particular spatial distribution, we need to ensure that Φ( ′, unit) can only apply to regions of the same shape and size. Hence, Φ must satisfy the condition ∀′[Φ( ′, unit) → CG(′, ) ], where CG is the relation of congruence (which in region-based Euclidean geometry is definable from parthood and sphericity primitives [ 34]). So, the complex predicate Φ(, ) must use spatial primitives (e.g. parthood and sphericity) to specify a particular shape of spatial relation (e.g. it consists of spheres arranged in a particular way) and the size of this configuration is fixed in relation to that of the particular sphere designated by . So when we name a particular sphere as unit and name a particular region , which satisfies Φ(, unit) , as being the region matter, we fix a unique association between the vocabulary of the theory and particular spatial regions.
Note that the coordinate anchoring shown in Fig. 1(a) is a special case of this kind of construction, where ‘unit’ is the disc centred on the origin and ‘matter’ can be taken to be the sum of all the four discs. Moreover, we could take the shape constructed in Fig. 1(b) as the denotation of ‘matter’ and (again taking ‘unit’ as the disc at the origin) it would not be so dificult to specify a predicate Φ such that Φ(matter, unit) would hold for this case, and Φ would also satisfy the given condition regarding congruence.
In view of the possibility of constructing a theory of space and matter of the form just described, we can understand the resolution of Tarski’s puzzle as follows. We cannot fix space without introducing some reference points or regions that specify a coordinate system. Geometers have, as one would expect, solved this problem to their own satisfaction by specifying a coordinate system with purely geometrical elements (designated points or spheres to indicate a unit length and the orientation of axes).
But such a formulation of geometry cannot serve as a grounding for interpreting material properties of the world, since the reference markers are just arbitrary points or regions in space with no observable characteristics to distinguish them from any other points or regions in space. Basing the coordinate system on the location of matter in space makes sense, not only as a practical means of establishing a coordinate system but also as an explanation of how a formal theory of material objects and processes can be constructed in relation to a geometric theory that only describes empty space.
The observations of the previous section indicate that we can treat matter as a primitive notion independent from space; and, as well as providing a reference frame it can also provide a foundation for the definition of classes of physical objects and their properties and relations. Matter can either be considered as a continuous substance or as consisting of atoms. The decisive shift to atomism becoming the dominant view came only in the early 20th century after Einstein’s 1905 paper on Brownian motion provided strong empirical evidence for atoms.
One way to ground a theory of physical reality and anchor it to space and time is simply to stipulate that there are one or more basic types of matter; and, at each point in time, each matter type occupies some particular region of space. This is essentially just an elaboration of the solution to Tarski’s puzzle given above and illustrated in Fig. 1.
Ontologies such as ChEBI [35] characterise chemical entities of a wide variety of types that are significant for chemists and particularly in relation to their role in biology. ChEBI also incorporates relationships of constituency that relate simple entities to complex ones, such as atoms to molecules. This is certainly very useful for a wide variety of applications where systematic organisation of information relating to chemistry and biology is important. However, ChEBI is largely Aristotelian in its content: it provides a classification of objects but (although it does indicate important constituency relationships) it does not explicitly define all its classes in terms of the structure and constituents of the members of those classes. Clearly doing that would be a huge and complex task. Nevertheless, there does not seem to be any reason why this would not be possible.
The following formulation of molecular structure and the class of molecules is only illustrative. It is strongly definitional in that it is constructed using formulae in which the only undefined primitive is the parthood relation, P(, ), interpreted as ‘Region is part of region ’. This presentation glosses over some significant complexities, in particular, the time dependence of certain predicates and relations. • O(, ) ≡ ∃ [ P(, ) ∧ P(, ) ] • ∀[ (sum(, ) = ) ↔ ∀ [ O(, ) ↔ ( O(, ) ∨ O(, ) ) ] • At() ≡ ∀[ P(, ) → = ]
An object is a collection of objects of type if and only if every part of overlaps with some object of type which is part of :
_Coll() ≡ def ∀[P(, ) → ∃[O(, ) ∧ () ∧ P(, )]]
An aggregate of entities of a given type with respect to a particular cohesion relation is a collection of entities of type , such that any two collections 1 and 2 whose sum is are such that we can find two entities of type that are respectively parts of 1 and 2, and which are related by the cohesion relation.
__Agg() ≡ def _Coll() ∧ ∀12[ (_Coll( 1) ∧ _Coll( 2) ∧ ( = sum(1, 2)) →
∃12[ ( 1) ∧ ( 2) ∧ P(1, 1) ∧ P(2, 2) ∧ ( 1, 2)]
So a material structure constituted by a collection of atoms joined by atomic bonds can be characterised by the predicate At_AtBond_Agg. The cohesion constraint means that it is not possible to divide the collection into two sub-collections, such that there is no atomic bond linking the sub-collections.
Max_() ≡
def () ∧ ∀ [ (() ∧ P(, )) → = ] So a molecule can be defined as a maximal aggregate of atoms that are joined by atomic bonds:
Mol() ≡ def Max_At_AtBond_Agg()
The atomic approach to matter can also be seen as a variant solution to Tarski’s puzzle, where we ifx space in relation to regions occupied by atomic or sub-atomic particles rather than types of matter distributed continuously over regions of space.
This paper has examined the definitional structure of formal ontologies with a particular focus on how material content can be incorporated into geometrical theories of space and time. Beginning from the model-theoretic foundations of definability, we reviewed key results including Padoa’s method, Beth’s theorem, and Tarski’s analysis of conceptual completeness. These results provide not only tools for assessing the definitional coherence of a theory but also impose subtle constraints on the way new concepts—such as matter—can be introduced.
A central puzzle identified by Tarski is that Euclidean geometry is conceptually complete: any extension by new concepts must, if the theory is categorical, result in those concepts being definable from the original vocabulary. At the same time, it appears that purely geometrical notions are insuficient to characterise matter and physical objects. To address this apparent contradiction, we proposed that matter can be introduced at the point where a coordinate system is anchored—thus breaking model symmetries and fixing reference in a way that makes definitional completeness compatible with the inclusion of physical content.
Two approaches to the ontological representation of matter were considered. A continuous theory posits that matter types occupy regions of space at each time, allowing geometric anchoring through structured spatial distributions. Alternatively, a discrete, atomistic theory enables definitions of molecules and chemical structures as aggregates of atoms constrained by cohesion relations. Both approaches are compatible with the formal machinery of definitional ontology and suggest routes to reconciling classical physical theories with foundational logical frameworks.
This investigation has shown that the puzzle posed by Tarski can be resolved without abandoning classical geometry or ontology. Instead, by carefully considering the point at which semantic content is anchored, we can see how even seemingly qualitative features of the physical world—such as the presence of objects and substances—can be accounted for within a rigorously definitional framework. This opens the way for more grounded and formally precise constructions of mid-level ontologies that remain tied to fundamental physical structure.
Looking forward, two main directions suggest themselves for extending this approach. The first is to build upwards from the foundational treatment of matter by defining specific classes of physical objects in terms of structured configurations of material regions, and thereby incrementally extend conceptual terminology to incorporate mid-level vocabularies closer to those used in ordinary language. The second is to introduce one or more temporal primitives into the core foundation theory. These could then be used to define high-level concepts characterising events and processes.
While physical theories are typically axiomatized in terms of the evolution of matter over time, the perspective ofered by our resolution of Tarski’s puzzle suggests that physical laws themselves might be grounded in an extended matter-based coordinate system, one that encompasses not just spatial anchoring but also temporal structure. If we begin with a system that only fixes the geometry at a single time point, it is likely that one would again encounter the limitation of conceptual completeness, and be unable to enrich the theory with concepts that capture dynamic change. This suggests that the temporal structure of the world and the laws that govern it should be integrated from the outset into the axiomatic treatment of matter, so that the propensity for its distribution to evolve over time is built directly into its meaning.
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